\(1 \mathrm{~g}=\ldots \ldots \ldots \ldots \ldots\) amu (a) \(6.02 \times 10^{23}\) (b) \(6.02 \times 10^{-23}\) (c) \(1.66 \times 10^{-27}\) (d) \(1.66 \times 10^{27}\)

An atomic mass unit (amu) is defined as 1/12 of the mass of a carbon-12 atom. The number of amu in 1 gram can be calculated using Avogadro's number, which is approximately \(6.022 \times 10^{23}\) atoms/mol. Since 1 mole of any substance contains Avogadro's number of atoms or molecules, and the molar mass of that substance in grams, we can use the molar mass and Avogadro's number to convert grams to amu.

First, we need to find the molar mass of the substance in question. Since we were only given the mass (1 gram) and not the substance, we have to make use of the mass-to-amu conversion: \[1\, \text{amu} = \frac{1\, \text{g}}{N_A}\] Where \(N_A\) represents Avogadro's number, which is approximately \(6.022 \times 10^{23} \, \text{atoms/mol}\). Now, we need to solve for the amu equivalent of 1 gram:

\[\text{amu} = \frac{1\, \text{g}}{6.022 \times 10^{23}\, \text{atoms/mol}} = 1.66 \times 10^{-24} \, \text{amu}\] Comparing this result to the options given in the exercise: (a) \(6.02 \times 10^{23}\) : It's Avogadro's number, but not the answer here. (b) \(6.02 \times 10^{-23}\) : Close to the reciprocal of Avogadro's number, but not the answer. (c) \(1.66 \times 10^{-27}\) : Not the correct conversion factor. (d) \(1.66 \times 10^{27}\) : Incorrect, since it is the reciprocal of our answer (1.66 × 10^{-24} amu). None of the options provided in the exercise exactly match our calculated answer (\(1.66 \times 10^{-24}\) amu). However, it is noteworthy that none of them is correct and the closest option (b) differs by a factor of 100. Please note that the number "\(1.66 \times 10^{-24}\) amu" that we found is only an approximation, and it is assumed that our calculations were aimed at a more accurate conversion value. Since there might be rounding errors, it would be wise to double-check the given exercise and the expected answer's precision.